Eigenvalue superposition for Toeplitz matrix-sequences with matrix order dependent symbols

The eigenvalues of Toeplitz matrices Tn(f) with a real-valued generating function f, satisfying some conditions and tracing out a simple loop over the interval [−π,π], are known to admit an asymptotic expansion with the form λj(Tn(f))=f(σj,n)+c1(σj,n)h+c2(σj,n)h2+O(h3), where h=1/(n+1), σj,n=πjh, and ck are some bounded coefficients depending only on f. The numerical results presented in the literature suggest that the effective conditions for the expansion to hold are weaker and reduce to a fixed smoothness and to having only two intervals of monotonicity over [−π,π]. In this article we investigate the superposition caused over this expansion, when considering the following linear combination λj(Tn(f0)+βn,1Tn(f1)+βn,2Tn(f2)), where βn,1,βn,2 are certain constants depending on n and the generating functions f0,f1,f2 are either simple loop or satisfy the weaker conditions mentioned before. We formally obtain an asymptotic expansion in this setting under simple-loop related assumptions, and we show numerically that there is much more to investigate, opening the door to linear in time algorithms for the computation of eigenvalues of large matrices of this type including a multilevel setting. The problem is of concrete interest, considering spectral features of matrices stemming from the numerical approximation of standard differential operators and distributed order fractional differential equations, via local methods such as Finite Differences, Finite Elements, and Isogeometric Analysis.